3.6.35 \(\int \frac {(1+x) (1+2 x+x^2)^5}{x^9} \, dx\)

Optimal. Leaf size=70 \[ -\frac {1}{8 x^8}-\frac {11}{7 x^7}-\frac {55}{6 x^6}-\frac {33}{x^5}-\frac {165}{2 x^4}+\frac {x^3}{3}-\frac {154}{x^3}+\frac {11 x^2}{2}-\frac {231}{x^2}+55 x-\frac {330}{x}+165 \log (x) \]

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Rubi [A]  time = 0.02, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {27, 43} \begin {gather*} \frac {x^3}{3}+\frac {11 x^2}{2}-\frac {231}{x^2}-\frac {154}{x^3}-\frac {165}{2 x^4}-\frac {33}{x^5}-\frac {55}{6 x^6}-\frac {11}{7 x^7}-\frac {1}{8 x^8}+55 x-\frac {330}{x}+165 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((1 + x)*(1 + 2*x + x^2)^5)/x^9,x]

[Out]

-1/(8*x^8) - 11/(7*x^7) - 55/(6*x^6) - 33/x^5 - 165/(2*x^4) - 154/x^3 - 231/x^2 - 330/x + 55*x + (11*x^2)/2 +
x^3/3 + 165*Log[x]

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin {align*} \int \frac {(1+x) \left (1+2 x+x^2\right )^5}{x^9} \, dx &=\int \frac {(1+x)^{11}}{x^9} \, dx\\ &=\int \left (55+\frac {1}{x^9}+\frac {11}{x^8}+\frac {55}{x^7}+\frac {165}{x^6}+\frac {330}{x^5}+\frac {462}{x^4}+\frac {462}{x^3}+\frac {330}{x^2}+\frac {165}{x}+11 x+x^2\right ) \, dx\\ &=-\frac {1}{8 x^8}-\frac {11}{7 x^7}-\frac {55}{6 x^6}-\frac {33}{x^5}-\frac {165}{2 x^4}-\frac {154}{x^3}-\frac {231}{x^2}-\frac {330}{x}+55 x+\frac {11 x^2}{2}+\frac {x^3}{3}+165 \log (x)\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 70, normalized size = 1.00 \begin {gather*} -\frac {1}{8 x^8}-\frac {11}{7 x^7}-\frac {55}{6 x^6}-\frac {33}{x^5}-\frac {165}{2 x^4}+\frac {x^3}{3}-\frac {154}{x^3}+\frac {11 x^2}{2}-\frac {231}{x^2}+55 x-\frac {330}{x}+165 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((1 + x)*(1 + 2*x + x^2)^5)/x^9,x]

[Out]

-1/8*1/x^8 - 11/(7*x^7) - 55/(6*x^6) - 33/x^5 - 165/(2*x^4) - 154/x^3 - 231/x^2 - 330/x + 55*x + (11*x^2)/2 +
x^3/3 + 165*Log[x]

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(1+x) \left (1+2 x+x^2\right )^5}{x^9} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((1 + x)*(1 + 2*x + x^2)^5)/x^9,x]

[Out]

IntegrateAlgebraic[((1 + x)*(1 + 2*x + x^2)^5)/x^9, x]

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fricas [A]  time = 0.39, size = 62, normalized size = 0.89 \begin {gather*} \frac {56 \, x^{11} + 924 \, x^{10} + 9240 \, x^{9} + 27720 \, x^{8} \log \relax (x) - 55440 \, x^{7} - 38808 \, x^{6} - 25872 \, x^{5} - 13860 \, x^{4} - 5544 \, x^{3} - 1540 \, x^{2} - 264 \, x - 21}{168 \, x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)*(x^2+2*x+1)^5/x^9,x, algorithm="fricas")

[Out]

1/168*(56*x^11 + 924*x^10 + 9240*x^9 + 27720*x^8*log(x) - 55440*x^7 - 38808*x^6 - 25872*x^5 - 13860*x^4 - 5544
*x^3 - 1540*x^2 - 264*x - 21)/x^8

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giac [A]  time = 0.15, size = 59, normalized size = 0.84 \begin {gather*} \frac {1}{3} \, x^{3} + \frac {11}{2} \, x^{2} + 55 \, x - \frac {55440 \, x^{7} + 38808 \, x^{6} + 25872 \, x^{5} + 13860 \, x^{4} + 5544 \, x^{3} + 1540 \, x^{2} + 264 \, x + 21}{168 \, x^{8}} + 165 \, \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)*(x^2+2*x+1)^5/x^9,x, algorithm="giac")

[Out]

1/3*x^3 + 11/2*x^2 + 55*x - 1/168*(55440*x^7 + 38808*x^6 + 25872*x^5 + 13860*x^4 + 5544*x^3 + 1540*x^2 + 264*x
+ 21)/x^8 + 165*log(abs(x))

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maple [A]  time = 0.04, size = 59, normalized size = 0.84 \begin {gather*} \frac {x^{3}}{3}+\frac {11 x^{2}}{2}+55 x +165 \ln \relax (x )-\frac {330}{x}-\frac {231}{x^{2}}-\frac {154}{x^{3}}-\frac {165}{2 x^{4}}-\frac {33}{x^{5}}-\frac {55}{6 x^{6}}-\frac {11}{7 x^{7}}-\frac {1}{8 x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x+1)*(x^2+2*x+1)^5/x^9,x)

[Out]

-1/8/x^8-11/7/x^7-55/6/x^6-33/x^5-165/2/x^4-154/x^3-231/x^2-330/x+55*x+11/2*x^2+1/3*x^3+165*ln(x)

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maxima [A]  time = 0.62, size = 58, normalized size = 0.83 \begin {gather*} \frac {1}{3} \, x^{3} + \frac {11}{2} \, x^{2} + 55 \, x - \frac {55440 \, x^{7} + 38808 \, x^{6} + 25872 \, x^{5} + 13860 \, x^{4} + 5544 \, x^{3} + 1540 \, x^{2} + 264 \, x + 21}{168 \, x^{8}} + 165 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)*(x^2+2*x+1)^5/x^9,x, algorithm="maxima")

[Out]

1/3*x^3 + 11/2*x^2 + 55*x - 1/168*(55440*x^7 + 38808*x^6 + 25872*x^5 + 13860*x^4 + 5544*x^3 + 1540*x^2 + 264*x
+ 21)/x^8 + 165*log(x)

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mupad [B]  time = 0.03, size = 58, normalized size = 0.83 \begin {gather*} 55\,x+165\,\ln \relax (x)-\frac {330\,x^7+231\,x^6+154\,x^5+\frac {165\,x^4}{2}+33\,x^3+\frac {55\,x^2}{6}+\frac {11\,x}{7}+\frac {1}{8}}{x^8}+\frac {11\,x^2}{2}+\frac {x^3}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x + 1)*(2*x + x^2 + 1)^5)/x^9,x)

[Out]

55*x + 165*log(x) - ((11*x)/7 + (55*x^2)/6 + 33*x^3 + (165*x^4)/2 + 154*x^5 + 231*x^6 + 330*x^7 + 1/8)/x^8 + (
11*x^2)/2 + x^3/3

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sympy [A]  time = 0.15, size = 61, normalized size = 0.87 \begin {gather*} \frac {x^{3}}{3} + \frac {11 x^{2}}{2} + 55 x + 165 \log {\relax (x )} + \frac {- 55440 x^{7} - 38808 x^{6} - 25872 x^{5} - 13860 x^{4} - 5544 x^{3} - 1540 x^{2} - 264 x - 21}{168 x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)*(x**2+2*x+1)**5/x**9,x)

[Out]

x**3/3 + 11*x**2/2 + 55*x + 165*log(x) + (-55440*x**7 - 38808*x**6 - 25872*x**5 - 13860*x**4 - 5544*x**3 - 154
0*x**2 - 264*x - 21)/(168*x**8)

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